There's oil underneath those rocks and water! The sale of an island to the government of the country which controls the island produced riots in the country with a contested claim.

Kuril Islands

Transfer of this volcano-intensive archipelago of 56 islands was effected by Yalta, but never accepted by the contesting country; major reason a truce ending WWII has never been signed by these two nations.

The Korean Peninula

These two countries are still at war, despite an armistice.

Western Sahara (give indigenous people seeking independence and country)

The Polisario Front has led an armed inurgency. In 1991 a peace proposal called for a referendum, but the country relocated settlers to the contested area and the armed campaigns continued.

Antarctica

Name any one of the countries that has a territorial claim on Antarctica, despite a treaty signed by 48 countries prohibiting such claims. Previously Nazi Germany had also claimed New Swabia.

'The Holy Land'

Any couching of the controversy will depend on your political view, but the modern struggle has waged from the creation of the Jewish state in 1948.

Northern part of the country on the southern coast of the Gulf of Aden

The former British colony announced its independence from the former Italian colony, but has remained unrecognized.

Taiwan (give the official names of the two countries)

After WWII, Taiwan reverted to China, but the Chinese government was overthrown by the People's Liberation Army of Mao Zedong. Chiang-Kai-shek set up the nationalist government here.

Crimea and Donbass

Despite one country having been assured of their security when they agreed to dismantle their nukes, and through a treaty with Yeltsin, the other country annexed the Crimea and is still fighting a war in Donbass to 'protect' people of their heritage.

Kashmir

In addition to the People's Republic of China, name the two other countries that share control of this contested territory. Boundaries had been set after WWII, but China never accepted them.

Cyprus

Give the name of the country, only recognized by Turkey, that has de facto rule over part of the island.